arXiv subjects

Tryphon T. Georgiou

Publications and source records attributed to Tryphon T. Georgiou.

At least 37 records · Page 2Linked to original sources

arXiv2024

Particle approximations of Wigner distributions for n arbitrary observables

A class of signed joint probability measures for n arbitrary quantum observables is derived and studied based on quasi-characteristic functions with symmetrized operator orderings of Margenau-Hill type. It is shown that the Wigner distribution associated with these observables can be rigorously approximated by such measures. These measures are given by affine combinations of Dirac delta distributions supported over the finite spectral range of the quantum observables and give the correct probability marginals when coarse-grained along any principal axis. We specialize to bivariate quasi-probability distributions for the spin measurements of spin-1/2 particles and derive their closed-form expressions. As a side result, we point out a connection between the convergence of these particle approximations and the Mehler-Heine theorem. Finally, we interpret the supports of these quasi-probability distributions in terms of repeated thought experiments.

quant-ph↗

arXiv2024

Wasserstein speed limits for Langevin systems

Physical systems transition between states with finite speed that is limited by energetic costs. In this work, we derive bounds on transition times for general Langevin systems that admit a decomposition into reversible and irreversible dynamics, in terms of the Wasserstein distance between states and the energetic costs associated with respective reversible and irreversible currents. For illustration we discuss Brownian particles subject to arbitrary forcing and an RLC circuit with time-varying inductor.

math-ph↗

arXiv2024

Refined bounds on energy harvesting from anisotropic fluctuations

We consider overdamped Brownian particles with two degrees of freedom (DoF) that are confined in a time-varying quadratic potential and are in simultaneous contact with heat baths of different temperatures along the respective DoF. The anisotropy in thermal fluctuations can be used to extract work by suitably manipulating the confining potential. The question of what the maximal amount of work that can be extracted is has been raised in recent work, and has been computed under the simplifying assumption that the entropy of the distribution of particles (thermodynamic states) remains constant throughout a thermodynamic cycle. Indeed, it was shown that the maximal amount of work that can be extracted amounts to solving an isoperimetric problem, where the 2-Wasserstein length traversed by thermodynamic states quantifies dissipation that can be traded off against an area integral that quantifies work drawn out of the thermal anisotropy. Here, we remove the simplifying assumption on constancy of entropy. We show that the work drawn can be computed similarly to the case where the entropy is kept constant while the dissipation can be reduced by suitably tilting the thermodynamic cycle in a thermodynamic space with one additional dimension. Optimal cycles can be locally approximated by solutions to an isoperimetric problem in a tilted lower-dimensional subspace.

cond-mat.stat-mech↗

arXiv2024

An excursion onto Schrödinger's bridges: Stochastic flows with spatio-temporal marginals

The purpose of the present work is to expand substantially the type of control and estimation problems that can be addressed following the paradigm of Schrödinger bridges, by incorporating termination (killing) of stochastic flows. Specifically, in the context of estimation, we seek the most likely evolution realizing measured spatio-temporal marginals of killed particles. In the context of control, we seek a suitable control action directing the killed process toward spatio-temporal probabilistic constraints. To this end, we derive a new Schrödinger system of coupled, in space and time, partial differential equations to construct the solution of the proposed problem. Further, we show that a Fortet-Sinkhorn type of algorithm is available to attain the associated bridge. A key feature of our framework is that the obtained bridge retains the Markovian structure in the prior process, and thereby, the corresponding controller takes the form of state feedback.

math.OC↗

arXiv2024

Network Learning with Directional Sign Patterns

Complex systems can be effectively modeled via graphs that encode networked interactions, where relations between entities or nodes are often quantified by signed edge weights, e.g., promotion/inhibition in gene regulatory networks, or encoding political of friendship differences in social networks. However, it is often the case that only an aggregate consequence of such edge weights that characterize relations may be directly observable, as in protein expression of in gene regulatory networks. Thus, learning edge weights poses a significant challenge that is further exacerbated for intricate and large-scale networks. In this article, we address a model problem to determine the strength of sign-indefinite relations that explain marginal distributions that constitute our data. To this end, we develop a paradigm akin to that of the Schrödinger bridge problem and an efficient Sinkhorn type algorithm (more properly, Schrödinger-Fortet-Sinkhorn algorithm) that allows fast convergence to parameters that minimize a relative entropy/likelihood criterion between the sought signed adjacency matrix and a prior. The formalism that we present represents a novel generalization of the earlier Schrödinger formalism in that marginal computations may incorporate weights that model directionality in underlying relations, and further, that it can be extended to high-order networks -- the Schrödinger-Fortet-Sinkhorn algorithm that we derive is applicable all the same and allows geometric convergence to a sought sign-indefinite adjacency matrix or tensor, for high-order networks. We demonstrate our framework with synthetic and real-world examples.

math.OC↗

arXiv2024

Inferring potential landscapes: A Schrödinger bridge approach to Maximum Caliber

Schrödinger bridges have emerged as an enabling framework for unveiling the stochastic dynamics of systems based on marginal observations at different points in time. The terminology "bridge'' refers to a probability law that suitably interpolates such marginals. The theory plays a pivotal role in a variety of contemporary developments in machine learning, stochastic control, thermodynamics, and biology, to name a few, impacting disciplines such as single-cell genomics, meteorology, and robotics. In this work, we extend Schrödinger's paradigm of bridges to account for integral constraints along paths, in a way akin to Maximum Caliber - a Maximum Entropy principle applied in a dynamic context. The Maximum Caliber principle has proven useful to infer the dynamics of complex systems e.g., that model gene circuits and protein folding. We unify these two problems via a maximum likelihood formulation to reconcile stochastic dynamics with ensemble-path data. A variety of data types can be encompassed, ranging from distribution moments to average currents along paths. The framework enables inference of time-varying potential landscapes that drive the process. The resulting forces can be interpreted as the optimal control that drives the system in a way that abides by specified integral constraints. Analogous results are presented in a discrete-time, discrete-space setting and specialized to steady-state dynamics. We finish by illustrating the practical applicability of the framework through paradigmatic examples, such as that of bit erasure or protein folding. In doing so, we highlight the strengths of the proposed framework, namely, the generality of the theory, the ease of computation, and the ability to interpret results in terms of system dynamics. This is in contrast to Maximum-Caliber problems where the focus is typically on updating a probability law on paths.

cond-mat.stat-mech↗

arXiv2023

Computational Optimal Transport and Filtering on Riemannian manifolds

In this paper we extend recent developments in computational optimal transport to the setting of Riemannian manifolds. In particular, we show how to learn optimal transport maps from samples that relate probability distributions defined on manifolds. Specializing these maps for sampling conditional probability distributions provides an ensemble approach for solving nonlinear filtering problems defined on such geometries. The proposed computational methodology is illustrated with examples of transport and nonlinear filtering on Lie groups, including the circle $S^1$, the special Euclidean group $SE(2)$, and the special orthogonal group $SO(3)$.

math.OC↗

arXiv2023

Data Assimilation for Sign-indefinite Priors: A generalization of Sinkhorn's algorithm

The purpose of this work is to develop a framework to calibrate signed datasets so as to be consistent with specified marginals by suitably extending the Schrödinger-Fortet-Sinkhorn paradigm. Specifically, we seek to revise sign-indefinite multi-dimensional arrays in a way that the updated values agree with specified marginals. Our approach follows the rationale in Schrödinger's problem, aimed at updating a "prior" probability measure to agree with marginal distributions. The celebrated Sinkhorn's algorithm (established earlier by R.\ Fortet) that solves Schrödinger's problem found early applications in calibrating contingency tables in statistics and, more recently, multi-marginal problems in machine learning and optimal transport. Herein, we postulate a sign-indefinite prior in the form of a multi-dimensional array, and propose an optimization problem to suitably update this prior to ensure consistency with given marginals. The resulting algorithm generalizes the Sinkhorn algorithm in that it amounts to iterative scaling of the entries of the array along different coordinate directions. The scaling is multiplicative but also, in contrast to Sinkhorn, inverse-multiplicative depending on the sign of the entries. Our algorithm reduces to the classical Sinkhorn algorithm when the entries of the prior are positive.

stat.ML↗

arXiv2023

Promotion/Inhibition Effects in Networks: A Model with Negative Probabilities

Biological networks often encapsulate promotion/inhibition as signed edge-weights of a graph. Nodes may correspond to genes assigned expression levels (mass) of respective proteins. The promotion/inhibition nature of co-expression between nodes is encoded in the sign of the corresponding entry of a sign-indefinite adjacency matrix, though the strength of such co-expression (i.e., the precise value of edge weights) cannot typically be directly measured. Herein we address the inverse problem to determine network edge-weights based on a sign-indefinite adjacency and expression levels at the nodes. While our motivation originates in gene networks, the framework applies to networks where promotion/inhibition dictates a stationary mass distribution at the nodes. In order to identify suitable edge-weights we adopt a framework of ``negative probabilities,'' advocated by P.\ Dirac and R.\ Feynman, and we set up a likelihood formalism to obtain values for the sought edge-weights. The proposed optimization problem can be solved via a generalization of the well-known Sinkhorn algorithm; in our setting the Sinkhorn-type ``diagonal scalings'' are multiplicative or inverse-multiplicative, depending on the sign of the respective entries in the adjacency matrix, with value computed as the positive root of a quadratic polynomial.

q-bio.MN↗

arXiv2023

A matching principle for power transfer in Stochastic Thermodynamics

Gradients in temperature and particle concentration fuel many processes in the physical and biological world. In the present work we study a thermodynamic engine powered by anisotropic thermal excitation (that may be due to e.g., a temperature gradient), and draw parallels with the well-known principle of impedance matching in circuit theory, where for maximal power transfer, the load voltage needs to be half of that of the supplying power source. We maximize power output of the thermodynamic engine at steady-state and show that the optimal reactive force is precise half of that supplied by the anisotropy.

eess.SY↗

arXiv2023

Control and estimation of multi-commodity network flow under aggregation

A paradigm put forth by E. Schrödinger in 1931/32, known as Schrödinger bridges, represents a formalism to pose and solve control and estimation problems seeking a perturbation from an initial control schedule (in the case of control), or from a prior probability law (in the case of estimation), sufficient to reconcile data in the form of marginal distributions and minimal in the sense of relative entropy to the prior. In the same spirit, we consider traffic-flow and apply a Schrödinger-type dictum, to perturb minimally with respect to a suitable relative entropy functional a prior schedule/law so as to reconcile the traffic flow with scarce aggregate distributions on families of indistinguishable individuals. Specifically, we consider the problem to regulate/estimate multi-commodity network flow rates based only on empirical distributions of commodities being transported (e.g., types of vehicles through a network, in motion) at two given times. Thus, building on Schrödinger's large deviation rationale, we develop a method to identify {\em the most likely flow rates (traffic flow)}, given prior information and aggregate observations. Our method further extends the Schrödinger bridge formalism to the multi-commodity setting, allowing commodities to exit or enter the flow field as well (e.g., vehicles to enter and stop and park) at any time. The behavior of entering or exiting the flow field, by commodities or vehicles, is modeled by a Markov chains with killing and creation states. Our method is illustrated with a numerical experiment.

math.OC↗

arXiv2023

Minimal Wasserstein Surfaces

In finite-dimensions, minimal surfaces that fill in the space delineated by closed curves and have minimal area arose naturally in classical physics in several contexts. No such concept seems readily available in infinite dimensions. The present work is motivated by the need for precisely such a concept that would allow natural coordinates for a surface with a boundary of a closed curve in the Wasserstein space of probability distributions (space of distributions with finite second moments). The need for such a concept arose recently in stochastic thermodynamics, where the Wasserstein length in the space of thermodynamic states quantifies dissipation while area integrals (albeit, presented in a special finite-dimensional parameter setting) relate to useful work being extracted. Our goal in this work is to introduce the concept of a minimal surface and explore options for a suitable construction. To this end, we introduce the basic mathematical problem and develop two alternative formulations. Specifically, we cast the problem as a two-parameter Benamou-Breiner type minimization problem and derive a minimal surface equation in the form of a set of two-parameter partial differential equations. Several explicit solutions for minimal surface equations in Wasserstein spaces are presented. These are cast in terms of covariance matrices in Gaussian distributions.

math.OC↗

arXiv2023

Optimal transport through a toll

We address the problem of optimal transport with a quadratic cost functional and a constraint on the flux through a constriction along the path. The constriction, conceptually represented by a toll station, limits the flow rate across. We provide a precise formulation which, in addition, is amenable to generalization in higher dimensions. We work out in detail the case of transport in one dimension by proving existence and uniqueness of solution. Under suitable regularity assumptions we give an explicit construction of the transport plan. Generalization of flux constraints to higher dimensions and possible extensions of the theory are discussed.

eess.SY↗

arXiv2023

Monge-Kantorovich Optimal Transport Through Constrictions and Flow-rate Constraints

We consider the problem to transport resources/mass while abiding by constraints on the flow through constrictions along their path between specified terminal distributions. Constrictions, conceptualized as toll stations at specified points, limit the flow rate across. We quantify flow-rate constraints via a bound on a sought probability density of the times that mass-elements cross toll stations and cast the transportation scheduling in a Kantorovich-type of formalism. Recent work by our team focused on the existence of Monge maps for similarly constrained transport minimizing average kinetic energy. The present formulation in this paper, besides being substantially more general, is cast as a (generalized) multi-marginal transport problem - a problem of considerable interest in modern-day machine learning literature and motivated extensive computational analyses. An enabling feature of our formalism is the representation of an average quadratic cost on the speed of transport as a convex constraint that involves crossing times.

eess.SY↗

arXiv2023

Minimal entropy production in the presence of anisotropic fluctuations

Anisotropy in temperature, chemical potential, or ion concentration, provides the fuel that feeds dynamical processes that sustain life. At the same time, anisotropy is a root cause of incurred losses manifested as entropy production. In this work we consider a rudimentary model of an overdamped stochastic thermodynamic system in an anisotropic temperature heat bath, and study minimum entropy production when driving the system between thermodynamic states in finite time. While entropy production in isotropic temperature environments can be expressed in terms of the length (in the Wasserstein-2 metric) traversed by the thermodynamic state of the system, anisotropy complicates substantially the mechanism of entropy production since, besides dissipation, seepage of energy between ambient anisotropic heat sources by way of the system dynamics is often a major contributing factor. A key result of the paper is to show that in the presence of anisotropy, minimization of entropy production can once again be expressed via a modified Optimal Mass Transport (OMT) problem. However, in contrast to the isotropic situation that leads to a classical OMT problem and a Wasserstein length, entropy production may not be identically zero when the thermodynamic state remains unchanged (unless one has control over non-conservative forces); this is due to the fact that maintaining a Non-Equilibrium Steady-State (NESS) incurs an intrinsic entropic cost that can be traced back to a seepage of heat between heat baths. As alluded to, NESSs represent hallmarks of life, since living matter by necessity operates far from equilibrium. Therefore, the question studied herein, to characterize minimal entropy production in anisotropic environments, appears of central importance in biological processes and on how such processes may have evolved to optimize for available usage of resources.

cond-mat.stat-mech↗

arXiv2022

Geometry of finite-time thermodynamic cycles with anisotropic thermal fluctuations

In contrast to the classical concept of a Carnot engine that alternates contact between heat baths of different temperatures, naturally occurring processes usually harvest energy from anisotropy, being exposed simultaneously to chemical and thermal fluctuations of different intensities. In these cases, the enabling mechanism responsible for transduction of energy is typically the presence of a non-equilibrium steady state (NESS). A suitable stochastic model for such a phenomenon is the Brownian gyrator -- a two-degree of freedom stochastically driven system that exchanges energy and heat with the environment. In the context of such a model we present, from a stochastic control perspective, a geometric view of the energy harvesting mechanism that entails a forced periodic trajectory of the system state on the thermodynamic manifold. Dissipation and work output are expressed accordingly as path integrals of a controlled process, and fundamental limitations on power and efficiency are expressed in geometric terms via a relationship to an isoperimetric problem. The theory is presented for high-order systems far from equilibrium and beyond the linear response regime.

eess.SY↗

arXiv2022

Minimal entropy production in anisotropic temperature fields

Anisotropy of temperature fields, chemical potentials and ion concentration gradients provide the fuel that feeds dynamical processes that sustain life. Dynamical flows in respective environments incur losses manifested as entropy production. In this work we consider a rudimentary model of an overdamped stochastic thermodynamic system in an anisotropic temperature heat bath, and analyze the problem to minimize entropy production while driving the system between thermodynamic states in finite time. It is noted that entropy production in a fully isotropic temperature field, can be expressed as the Wasserstein-2 length of the path traversed by the thermodynamic state of the system. In the presence of an anisotropic temperature field, the mechanism of entropy production is substantially more complicated as, besides dissipation, it entails seepage of energy between the ambient heat sources by way of the system dynamics. We show that, in this case, the entropy production can be expressed as the solution of a suitably constrained and generalized Optimal Mass Transport (OMT) problem. In contrast to the situation in standard OMT, entropy production may not be identically zero, even when the thermodynamic state remains unchanged. Physically, this is due to the fact that maintaining a Non-Equilibrium Steady State (NESS), incurs an intrinsic entropic cost. As already noted, NESSs are the hallmark of life and living systems by necessity operate away from equilibrium. Thus our problem of minimizing entropy production appears of central importance in understanding biological processes, such as molecular motors and motor proteins, and on how such processes may have evolved to optimize for available usage of resources.

cond-mat.stat-mech↗

arXiv2022

Inertialess Gyrating Engines

A typical model for a gyrating engine consists of an inertial wheel powered by an energy source that generates an angle-dependent torque. Examples of such engines include a pendulum with an externally applied torque, Stirling engines, and the Brownian gyrating engine. Variations in the torque are averaged out by the inertia of the system to produce limit cycle oscillations. While torque generating mechanisms are also ubiquitous in the biological world, where they typically feed on chemical gradients, inertia is not a property that one naturally associates with such processes. In the present work, seeking ways to dispense of the need for inertial effects, we study an inertia-less concept where the combined effect of coupled torque-producing components averages out variations in the ambient potential and helps overcome dissipative forces to allow sustained operation for vanishingly small inertia. We exemplify this inertia-less concept through analysis of two of the aforementioned engines, the Stirling engine and the Brownian gyrating engine. An analogous principle may be sought in biomolecular processes as well as in modern-day technological engines, where for the latter, the coupled torque-producing components reduce vibrations that stem from the variability of the generated torque.

cond-mat.stat-mech↗